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A combined experimental/numerical investigation on hygrothermal aging of fiber-reinforced composites

Rocha, I.B.C.M.; van der Meer, F.P.; Raijmaekers, S.; Lahuerta, F.; Nijssen, R.P.L.; Mikkelsen, LarsPilgaard; Sluys, L.J.

Published in:European Journal of Mechanics A - Solids

Link to article, DOI:10.1016/j.euromechsol.2018.10.003

Publication date:2018

Document VersionEarly version, also known as pre-print

Link back to DTU Orbit

Citation (APA):Rocha, I. B. C. M., van der Meer, F. P., Raijmaekers, S., Lahuerta, F., Nijssen, R. P. L., Mikkelsen, L. P., &Sluys, L. J. (2018). A combined experimental/numerical investigation on hygrothermal aging of fiber-reinforcedcomposites. European Journal of Mechanics A - Solids, 73, 407-419.https://doi.org/10.1016/j.euromechsol.2018.10.003

A combined experimental/numerical investigation on

hygrothermal aging of fiber-reinforced composites

I. B. C. M. Rochaa,b,∗, F. P. van der Meerb, S. Raijmaekersa, F. Lahuertaa, R. P. L.

Nijssena, L. P. Mikkelsenc, L. J. Sluysb

aKnowledge Centre WMC, Kluisgat 5, 1771MV Wieringerwerf, The NetherlandsbDelft University of Technology, Faculty of Civil Engineering and Geosciences, P.O. Box 5048, 2600GA

Delft, The NetherlandscTechnical University of Denmark, Department of Wind Energy, Composites and Materials Mechanics,

DTU Risø Campus, 4000 Roskilde, Denmark

Abstract

This work investigates hygrothermal aging degradation of unidirectional glass/epoxy

composite specimens through a combination of experiments and numerical model-

ing. Aging is performed through immersion in demineralized water. Interlaminar

shear testes are performed after multiple conditioning times and after single immer-

sion/redrying cycles. Degradation of the fiber-matrix interface is estimated using single-

fiber fragmentation tests and reverse modeling combining analytical and numerical

models. A fractographic analysis of specimens aged at 50 C and 65 C is performed

through X-ray computed tomography. The aging process is modeled using a numeri-

cal framework combining a diffusion analysis with a concurrent multiscale model with

embedded hyper-reduced micromodels. At the microscale, a pressure-dependent vis-

coelastic/viscoplastic model with damage is used for the resin and fiber-matrix debond-

ing is modeled with a cohesive-zone model including friction. A comparison between

numerical and experimental results is performed.

Keywords: Fiber-reinforced composites, multiscale analysis, reduced-order modeling

∗Corresponding author. Tel. +31 0227 504928

Email address: i.barcelos@wmc.eu (I. B. C. M. Rocha)

Preprint submitted to Elsevier September 12, 2018

1. Introduction

Fiber-reinforced composite materials feature an intrinsically multiscale behavior.

Owing to their complex microstructures, predicting failure in composite materials re-

quires knowledge on a number of microscopic failure mechanisms and on their highly

nonlinear interactions leading up to macroscopic failure [1–3].

Complexity is further increased when the interaction between the material and its

service environment is taken into account. In the particular case of hygrothermal ag-

ing, a combination of high temperatures and moisture ingression, processes such as

differential swelling, plasticization and hydrolysis can severely impact the mechani-

cal performance of the composite material [4–6]. Besides affecting each microscopic

material component differently, these degradation processes may act on different time

scales and be either reversible or irreversible [7].

Research on hygrothermal aging of fiber-reinforced composites has often focused

on macroscopic experiments which, although providing valuable information on ma-

terial degradation at scales closer to the ones used in structural design, offer limited

insight in the underlying microscopic aging mechanisms [8–11]. A number of authors

[1, 12, 13] attempt to further the understanding of these mechanisms through a com-

bination of micromechanical tests, novel micro- and nanoscale observation techniques

and high-fidelity numerical modeling, leading to more realistic predictions of compos-

ite material durability.

Micromechanical modeling allows for fibers, matrix and interfaces to be explic-

itly modeled and specialized constitutive models to be employed for each of them.

This enables capturing complex material behavior such as viscous effects in the resin

[14, 15] and debonding with friction at interfaces [1, 16]. Macroscopic behavior can in

turn be obtained through a bottom-up numerical homogenization procedure [1, 3] after

which the micromodel is substituted by a calibrated mesoscale model. Alternatively,

both scales can be treated concurrently through computational homogenization (FE2)

[17–19]. This approach allows for realistic modeling of macroscopic structures with

non-uniform degradation states without any mesoscale constitutive assumptions. How-

ever, as FE2 involves the concurrent modeling of multiple microscopic domains, gains

2

in fidelity and generality come at the cost of computational efficiency. In order to cir-

cumvent this drawback, reduced-order modeling techniques can be used to accelerate

the computation of the micromodels [20–22].

This paper presents a combined experimental and numerical investigation on hy-

grothermal aging of a glass/epoxy composite system. The work builds upon previous

experimental [7] and numerical [18] studies performed by the authors, combining them

and including a number of relevant improvements. For the experimental part, a new set

of mechanical tests is performed on purely unidirectional short-beam specimens aged

in demineralized water at 50 C for different durations in order to track the evolution of

material degradation, including tests on samples redried after having been both partially

and fully saturated. The macroscopic experimental investigation is complemented by

single-fiber fragmentation tests in order to estimate the fracture properties of the fiber-

matrix interface before and after aging. Furthermore, the fractographic analysis of aged

specimens presented in [7] is extended with a set of observations in specimens aged at

two different temperatures (50 C and 65 C) through X-ray 3D computed tomography.

For the numerical study, the multiscale and multiphysics numerical framework pro-

posed in [18] is improved by including a new viscoelastic/viscoplastic/damage consti-

tutive model for the resin [23] and a cohesive-zone model with friction for the fiber-

matrix interfaces. Furthermore, the full-order microscopic boundary value problem

embedded at each macroscopic material point is substituted by a hyper-reduced model

constructed using two different model order reduction techniques [24]. The modified

framework is used in an attempt to reproduce the experimentally obtained results on

short-beam specimens before and after aging.

2. Mechanical tests

In this section, the effects of hygrothermal aging in a glass/epoxy material system

are investigated through mechanical tests. More specifically, this work focuses on me-

chanical performance degradation of composite specimens subjected to interlaminar

shear and of the fiber-matrix interface adhesion as measured by single-fiber fragmenta-

tion tests (SFFT). The results obtained from the micromechanical tests are used as input

3

for the multiscale modeling framework of Section 4 while results from the macroscopic

tests are used to assess model performance.

2.1. Manufacturing and conditioning

The material system used in this work is a combination of the EPIKOTE RIMR

135/EPIKURE RIMH 1366 epoxy resin reinforced with unidirectional (UD) E-Glass

fiber fabrics composed of PPG Hybon 2002 fiber rovings. In order to obtain purely

UD laminates, the 90° stability roving layers originally included in the commercial UD

fabric were manually removed.

Dog-bone shaped single-fiber fragmentation specimens with 16mm gauge length,

2mm gauge width, 6.45mm tab width and 2mm thickness were manufactured by

extracting single fibers from a fabric and positioning them in latex molds into which

resin was poured and cured. For interlaminar shear strength (ILSS) tests, a 3-ply 320

mm× 320mm× 2.15mm panel was manufactured through vacuum infusion molding

and short-beams with 21.5mm length, 10.75mm width and 2.15mm thickness (ISO

14130 [25]) were cut from it using a CNC milling machine. The curing cycle for both

specimen types consisted of 2 h at 30 C, 5 h at 50 C and 10 h at 70 C.

After manufacturing, the specimens were kept in an evacuated desiccator at 50 C

and periodically weighed until a moisture-free state was reached. Sets of specimens

were tested in this dry reference state. The remaining specimens were immersed in

demineralized water at 50 C for different durations before being removed for test-

ing. ILSS specimens were immersed for periods varying from 250 h to 2000 h, dur-

ing which water uptake measurements through weighing were periodically conducted.

Single-fiber fragmentation specimens were immersed until saturation (approximately

500 h), after which weight measurements performed on two consecutive days were

used to confirm that saturation had been reached. Finally, sets of ILSS samples im-

mersed for 500 h and 1000 h were redried in the evacuated desiccator before being

tested. Weight measurements performed on two consecutive days were used to con-

firm that a stable dry state had been reached.

4

2.2. Testing

After conditioning, 10 single-fiber fragmentation specimens (5 dry, 5 saturated)

were tested in an MTS test frame with 1 kN load cell using a custom tensile fixture suit-

able for small dog-bone specimens. The specimen was tested in displacement control

at a rate of 0.5mmmin−1 until failure. A microscope camera equipped with polarized

light filters was used to record the development of fiber breaks.

Sets of 10 ILSS specimens for each condition were tested in three-point bending

at 1mmmin−1. The bending span was fixed at 11.4mm and steel cylinders with

diameters measuring 3.14mm and 6mm were used as supports and loading nose, re-

spectively.

2.3. Results and discussion

2.3.1. Single-fiber fragmentation tests

An attempt is made at estimating the fiber-matrix interface properties through a

combination of optic measurements around broken fiber fragments, the recent shear-

lag model proposed by Sørensen [16] and a finite element model of the fragmentation

process. Interface decohesion is modeled through a cohesive zone model with friction

(Section 4.1.2) and is characterized by a decohesion strength Xsh, a fracture toughness

GIIc and a friction coefficient µ.

Fig. 1a shows a zoomed-in view of a single fiber break during a fragmentation

test. Three quantities of interest are identified: The debonded length ld, which delimits

the region where interface adhesion is completely lost and only a constant friction

stress between fiber and matrix remains, the length lcz of the cohesive zone in which

debonding is taking place [26], and the distance lp between the brightest light spots in

the fragment, which can be reliably measured even at lower magnifications and used to

derive ld.

For each fiber break, the debonded length ld is plotted against the strain of the

gauge section of the dog-bone ε, measured through videoextensometry with a 10mm

gauge length. A straight line is then fitted through the data points:

ε = ε0 +d ε

d ( ld/rf)

ldrf

(1)

5

where ε0 is an intercept parameter and rf is the fiber radius. Care is taken to select

fragments which grow with no other fragments nearby and for strain values before

plastic localization occurs, two situations that violate assumptions made in the shear-

lag model for data reduction [16]. Assuming a constant friction stress τsh along the

debonded zone, the shear-lag model allows to evaluate its value from measured strain

and debond length as:

τsh =1

2

d ε

d ( ld/rf)Ef (2)

where Ef is the Young’s modulus of the fiber. In order to compute the fracture tough-

ness, a residual strain εres used to compute the strain at the fiber surface is estimated

from the combined actions of curing (shrinking) and water swelling (expansion). As-

suming that resin hardening takes place during the 50 C curing step and that chemical

shrinkage due to crosslinking reactions is negligible for the present multi-step curing

cycle [27], a differential strain εdryres = 0.0015 is obtained. For saturated specimens, the

swelling contribution is subtracted leading to εwetres = −0.0049. With these values, the

fracture toughness is computed as [16]:

GIIc =1

4Efrf (ε0 − εres)

2(3)

With values for τsh, GIIc and the cohesive zone length lcz, the interface strength

Xsh can be estimated through a finite element model with a one-dimensional fiber

with slip degrees of freedom embedded in a periodic resin slice (the reader is referred

to [28] for details on the model formulation). From Fig. 1b, features with the same

shapes as the ones seen through birefringence in the experiments can be observed. The

cohesive strength Xsh used as input in the model has a direct influence on the resultant

cohesive zone length lcz. This means that an estimate for Xsh can be obtained by fixng

the other parameters that also influence lcz, namely the frictional stress τsh and the

fracture energy GIIc, and adjusting the input strength until the numerical lcz matches

the experimentally measured one. Since neither the analytical nor the numerical model

take radial stresses into account, it is not possible to estimate the friction coefficient µ.

For the present, the value µ = 0.4, found by Naya et al. [1] to give the best fit with

experimental data for carbon/epoxy composites, is adopted.

6

2ldlcz

lp

lcz 2ld

(a) Experiment (b) Simulation

Figure 1: A fiber break as observed during an experiment and through numerical simulation, showing the

quantities of interest for property estimation.

Measurement results are presented in Figs. 2 and 3, showing adimensionalized

debonded lengths (ld/rf ) versus fiber strain (ε − εres) for 4 dry fragments and 12 wet

fragments. In the plots, the different colors and point markers represent measurements

on individual fragments. The reason for the lower number of dry fragments stems from

the relatively low failure strain of the present epoxy resin, which caused global failure

to happen in many of the tested specimens before useful debond measurements could

be made. For saturated specimens, in contrast, plasticization and differential swelling

promote earlier fragment development.

The averages and standard deviations of the obtained properties are shown in Ta-

ble 1, computed with lcz = 0.15mm, considered constant throughout the test. Due to

the indirect nature of the estimation procedure, a lack of literature consensus on the def-

inition of a debonded region with constant shear stress [16, 26], uncertainties related to

the amount of thermal and chemical residual strains and the large scatter observed be-

tween fragments, these values only provide a rough estimate of interface performance.

Further experiments and model development are therefore necessary. Nevertheless, re-

sults indicate loss of friction and degraded adhesion properties after aging, on average.

7

0 5 10 15

2.5

3

3.5

4

4.5

Debonded length (ld/rf ) [-]

Fib

erst

rain

(ε−

ε res)

[%]

Figure 2: SFFT results for dry specimens.

0 5 10 15 20

2

2.5

3

3.5

Debonded length (ld/rf ) [-]

Fib

erst

rain

(ε−ε r

es)

[%]

Figure 3: SFFT results for saturated specimens.

8

Dry Saturated

τsh [MPa] 46.27± 16.37 27.31± 13.22

GIIc [N/mm] 0.093± 0.056 0.067± 0.031

Xsh [MPa] 52.0 30.0

Table 1: Interface properties estimated through single-fiber fragmentation tests.

2.3.2. Interlaminar shear tests

For the macromechanical part of this study, hygrothermal degradation is measured

in unidirectional short beams tested in three-point bending. A measure of the transverse

shear stress at the center of the short-beam specimens is obtained from the force signal

F of the test frame [25]:

τILSS =3

4

F

bh(4)

where b and h are the specimen width and thickness, respectively, measured after con-

ditioning. Fig. 4 plots the thus computed shear stress versus crosshead displacement

for different aging times. To avoid clutter, only one representative specimen for each

condition is shown. Average values of the maximum attained stress (interlaminar shear

strength) for each condition are presented in Table 2.

In order to relate changes in strength to the water concentration in the specimens,

the water uptake is computed as:

w%(t) = 100m(t)−mdry

mdry

(5)

where m(t) is the specimen mass at time t and mdry is a reference mass measured

before aging. Water uptake and average interlaminar shear strength values (wet and

redried) are plotted against time in Fig. 5. It is important to note that unsaturated sam-

ples have a non-uniform water concentration field. The associated w% values for these

samples are therefore the volume averages of the concentration and do not represent

the exact amount of water at material regions where interlaminar failure occurs.

The results point to a gradual strength decrease as the specimens absorb water, with

up to 50% degradation after 1000 h, a point in time when the diffusion process subsides

9

0 0.2 0.4 0.6 0.80

20

40

60

Displacement [mm]

Sh

ear

stre

ss(τ

ILSS)

[MPa

]

Unaged Wet 250h

Wet 500h Wet 1000h

Wet 1500h Wet 2000h

Dry 500h Dry 1000h

Figure 4: Interlaminar shear strength (ILSS) results.

and the specimen is saturated with water. A clear correlation can therefore be identified

between the amount of water in the specimen and its interlaminar shear performance.

After reaching a stable value at saturation, the strength and uptake remain approxi-

mately constant between 1000 h and 2000 h, suggesting that long-term chemical degra-

dation of the interface [7] is negligible for the short aging durations considered in this

work. Nevertheless, a small part of the degradation is irreversible, as evidenced by the

redried tests performed after 500 h and 1000 h of immersion. The numerical models

of Section 4 will help elucidate if this permanent degradation arises from swelling-

induced microscopic failure events or if it is instead caused by a fast chemical degra-

dation process taking place at the same time scale as that of the diffusion phenomenon.

Unaged 250h 500h 1000h 1500h 2000h

Tested wet

τmaxILS [MPa] - 39.4± 1.0 34.4± 2.6 28.7± 0.9 29.5± 1.3 27.9± 0.9

Tested dry

τmaxILS [MPa] 56.7± 0.8 - 51.6± 0.9 51.8± 0.9 - -

Table 2: ILSS values for short-beam specimens.

10

0 500 1000 1500 2000

30

40

50

60

Immersion time [h]

Pea

kst

ress

(τmax

ILSS)

[MPa

]

Wet

Redried

0

0.2

0.4

0.6

0.8

1

1.2

Wat

erU

pta

ke

[%]

Uptake

Figure 5: ILSS peak stresses for wet and redried specimens and water uptake for different immersion times.

3. X-Ray computed tomography

Results from mechanical tests point to the occurrence of material damage after hy-

grothermal aging. It is therefore interesting to employ microscopic observation tech-

niques in an attempt to observe degradation events in specimens after aging but before

being mechanically tested.

3.1. Conditioning

For this part of the study, one additional short-beam specimen was conditioned

at 50 C for the longer period of 5000 h in order to investigate both short- and long-

term hygrothermal degradation. Furthermore, a specimen was conditioned at 65 C for

500 h so that degradation at two different temperatures can be compared. Observations

on an unaged specimen are also performed for comparison.

3.2. Scanning

Three-dimensional X-ray computed tomography scans of the specimens are per-

formed using a Zeiss Xradia Versa 520 scanner. Strip specimens measuring 2.2mm× 2.15

mm× 21.5mm are cut from the short-beams and glued to an aluminum pin which is

then attached to a special mount and positioned in the scanning chamber between an

11

Cut Mount

ROI

ILS specimen Pin mount Scanning chamber

Figure 6: Specimen preparation and scanning procedure, showing the Region of Interest (ROI) for the scans.

X-ray source and a detector (Fig. 6). A motorized x-y-z stage is used to align the scan-

ning Region of Interest (ROI) with the source and suitable source-sample and sample-

detector distances are chosen in order to obtain the desired scanning Field of View

(FoV). During the scans, tomographic projections of the ROI are taken as the sample

holder is gradually rotated. As a post-processing step, the projections are combined to

form three-dimensional reconstructions of the scanned volumes1.

As scanning resolution is inversely proportional to the size of the scanned volume

but the study involves searching for relatively small features (e.g. interfacial debonding

cracks on single fibers), a sequential scanning strategy is adopted. First, a large field of

view (LFoV) scan is performed and regions of interest close to the center and surface

of the specimens are chosen for small field of view scans (SFoV). Parameters for both

scan types are shown in Table 3.

3.3. Results and discussion

Fig. 7 shows the reconstructed cylindrical volume resulting from the LFoV scan of

an unaged specimen. The locations of the fiber bundles and resin rich regions along

1For the complete scan data, the reader is referred to [29]. The reader is also encouraged to scan the QR

codes embedded in the figures for videos of the reconstructed volumes.

12

Parameter LFoV SFoV

Magnification [-] 4.0x 20.0x

Beam voltage [keV] 30 45

Beam power [W] 2 3.5

Exposure time [s] 13-18 40

No. of projections [-] 5201-5801 3801

Field of view [µm] 2500 500

Scanning time [h/scan] 32 45

Table 3: Scanning parameters for small and large field of view scans.

the three plies that make up the specimen can be clearly identified. Fig. 8 shows SFoV

scans of the same specimen. No visible failure can be identified for this condition.

More interesting features are observed on the specimen aged at 65 C. The LFoV

scan of Fig. 9 shows a region of extensive interface debonding close to the specimen

surface. By moving along the specimen length (z-axis), it is possible to observe the

crack and its associated fracture process zone. Similar failure loci can also be observed

at multiple other points close to the specimen surface. The SFoV scan of Fig. 10a shows

one of these locations, with interface debonding both in isolated fibers and propagat-

ing among groups of fibers. In contrast, a zoomed-in scan close to the center of the

specimen reveals only barely visible debonding cracks (Fig. 10b).

For the specimen aged at 50 C, only minor failure events can be observed (Fig. 11).

Material degradation is therefore markedly worse for specimens aged at 65 C. Further-

more, the new spaces formed by crack opening promote additional water uptake, with

a moisture content of 1.5% measured after 500 h of immersion at 65 C, higher than

the observed saturation level of 1.2% for 50 C specimens (Fig. 5). Similar behavior

has been observed in other studies [4, 7, 9, 30].

Based on their proximity to the surface and on the large openings between crack

faces, it can be hypothesized that such debonding cracks, initiated through differential

swelling, propagate aided by an osmotic process that leads to accelerated water uptake

13

Fiber bundle Resin rich regions between bundles

y

x

z

Ply

stack

ing d

irection (th

ickness)

Figure 7: Reconstructed scanned volume of a dry specimen (LFoV).

(a) Specimen surface (b) Specimen center

Figure 8: Small field of view scans of a reference specimen.

14

x

y

Specimen length

Figure 9: Large field of view scan of a specimen aged at 65 C showing a large debonding crack.

and hydrostatic pressure between crack faces. This is similar to the behavior reported

for glass/polyester composites by Gautier et al. [4] and would imply that hydrolytic

chemical reactions take place involving either the interface sizing or the glass fibers,

creating leachates that drive the ensuing osmosis.

Activation of such osmotic mechanism would therefore depend on the differences

in speed between diffusion, chemical reaction and leaching. For specimens aged at

50 C, it is reasonable to suppose that diffusion and reaction are slow enough to allow

for reaction products to be leached before significant osmosis takes place. Neverthe-

less, hydrolytic reactions would permanently degrade interface performance, which

corresponds with the measured strength loss in the redried specimens of Section 2.3.2,

and cause loss of material through leaching, as reported by the authors in [7].

4. Numerical modeling

The aging process followed by mechanical testing is numerically simulated using

the Finite Element Method in order to reproduce the experimentally observed material

degradation. Since aging affects each material constituent differently, a micromechan-

ical modeling strategy is adopted. In order to realistically simulate tests on specimens

with non-homogeneous water concentration fields, a multiscale approach is used to

model both micro and macroscales concurrently. In this section, the resultant numeri-

cal framework is briefly presented in order to keep the study self-contained. For more

15

(a) Specimen surface (b) Specimen center

Figure 10: Small field of view scans of a specimen aged at 65 C, with cracks marked in red.

(a) Specimen surface (b) Specimen center

Figure 11: Small field of view scans of a specimen aged at 50 C, with small debonding cracks marked in

red.

16

detailed formulations, the interested reader will be referred to additional literature on

the subject.

4.1. Microscopic material models

4.1.1. Viscoelastic-Viscoplastic-Damage epoxy

An epoxy model with viscoelastic, viscoplastic and damage components is adopted

in order to take into account rate-dependent plasticity and damage activation. The

present development highlights the main ingredients of a model recently proposed by

the authors [23] with a slightly modified damage model inspired by the work of Arefi

et al. [31].

The model is based on an additive strain decomposition in elastic, plastic, thermal

and swelling parts:

ε = εe + ε

p + εth + ε

sw (6)

where the thermal and swelling contributions are given by:

εth = αt (T − T0) I ε

sw = αswcI (7)

with αt and αsw being the thermal and swelling expansion coefficients, respectively,

I being the identity matrix, c being the water concentration and T and T0 the actual

temperature and the temperature at which resin hardening took place, respectively.

The elastic strains are used to compute the stresses by integrating the complete

strain time history through the use of a time variable t:

σ(t) = D∞εe(t) +

∫ t

0

Dve(t− t)∂εe(t)

∂tdt (8)

The stresses are composed of an inviscid contribution related to the long-term stiffness

D∞ and a viscous contribution driven by the time-dependent viscoelastic stiffness Dve.

This viscous contribution is represented by a Prony series of bulk and shear stiffness

elements arranged in parallel, with stiffnesses Ku and Gv and relaxation times ku and

gv , respectively.

Plastic strain develops when a pressure-dependent paraboloidal yield surface is

reached:

fp(σ, εpeq) = 6J2 + 2I1

(σc(ε

peq)− σt(ε

peq))− 2σc(ε

peq)σt(ε

peq) (9)

17

where J2 and I1 are the second invariant of the deviatoric stress tensor and the first

invariant of the stress tensor, respectively, and σc and σt are the yield stresses in com-

pression and tension, respectively. Hardening is taken into account through the depen-

dency of the yield stresses on the equivalent plastic strain εpeq. The non-associative

plastic flow ∆εp is dictated by the plastic multiplier γ which in turn evolves in a vis-

cous manner:

∆εp = ∆γ

(3S+

2

9αI1I

)∆γ =

∆tηp

(fp

σ0t σ

0c

)mp

, iffp > 0

0, iffp ≤ 0

(10)

where S is the deviatoric stress tensor, α is a factor related to the plastic Poisson’s ratio

νp, ηp and mp are the viscoplastic modulus and exponent, respectively, σ0t and σ0

c are

the initial yield stresses, and ∆t is a time step.

A continuum damage model is adopted in order to model resin fracture. A damage

variable dm is adopted and the damaged stresses are computed as:

σ = DsH0σ (11)

where σ and H0 are the stresses and compliance matrix computed for the pristine

material (i.e. with dm = 0) and Ds is a secant stiffness matrix written as:

Ds =

κ β β 0 0 0

β κ β 0 0 0

β β κ 0 0 0

0 0 0 G(1− dm) 0 0

0 0 0 0 G(1− dm) 0

0 0 0 0 0 G(1− dm)

(12)

and the factors κ and β are given by:

κ =E(1− dm)(1− ν(1− dm))

(1 + ν(1− dm))(1− 2ν(1− dm))β =

Eν(1− dm)2

(1 + ν(1− dm))(1− 2ν(1− dm))(13)

with G being the viscoelastic shear modulus and E and ν the viscoelastic Young’s

modulus and Poisson’s ratio, computed as:

E = 2G (1 + ν) ν =3K − 2G

2G+ 6K(14)

18

This definition of the secant stiffness deviates from the original model proposed by the

authors [23] by also applying a degradation to ν. This makes the model more suitable

for use in dense finite element meshes by avoiding spurious hardening on damaged

elements constrained by bands of elements undergoing elastic unloading.

A pressure-dependent paraboloidal fracture surface is adopted:

fd(σ, r) =3J2XcXt

+I1(Xc −Xt)

XcXt

− r ⇒ fd(σ, r) = Λ− r (15)

where r is an internal variable which controls damage evolution and Xc = Xc(Υ) and

Xt = Xt(Υ) are the fracture strengths in compression and tension, respectively, which

are functions of the cumulative dissipated energy Υ = Υve + Υvp [23]. Through a

shrink in the fracture surfaces as energy dissipates, rate-dependent damage activation

is taken into account.

At each time step, the variable r is explicitly updated as:

rtn+1= max

1, max

0≤t≤tn+1

Λt

(16)

and the damage variable is computed based on the linear softening law proposed by

Arefi et al. [31]:

dm =

εf(εeq − ε0)

εeq(εf − ε0), εeq ≤ εf

1, εeq > εf

(17)

where the strains at the beginning (ε0) and end (εf ) of the softening regime are deter-

mined based on an uniaxial tensile test:

ε0 = Xt εf =2GcE

leXt

(18)

with Gc being the resin fracture toughness and le the characteristic finite element length

[32]. Finally, εeq is a scalar measure of the strain state:

εeq =−q +

√q2 − 4pu

2p(19)

19

and the parameters q, p and u are given by:

q = 2a(1− a)bεf + (1 + 2a)c u = b(aεf)2 − 2acεf − rXtXc

p = b(1− a)2 a =νXt

E(εf − ε0)

b =

(E

1 + ν

)2

c =

(E

1− 2ν

)(Xc −Xt)

(20)

4.1.2. Cohesive interfaces with friction

Fiber-matrix interface debonding is modeled using interface elements with soften-

ing behavior dictated by a cohesive-zone model. The model used in this work is based

on the one presented in [18] but is modified to include a friction component, following

the formulation by Alfano and Sacco [33]. The displacement jump is additively split

into elastic and friction contributions:

JuK = JuKe+ JuK

µ(21)

and a friction traction contribution tµ is added to the original cohesive traction:

t = (1− di)KJuK + ditµ (22)

where K is an initial stiffness, tµ = K (JuK − JuKµ), and di is a damage variable

which represents the degree of interface debonding.

The evolution of di with the total displacement jump JuK remains unchanged from

the original model [18] and will be omitted for compactness. Friction evolves in a way

analogous to non-associative plasticity. The displacement jump JuKµ

is updated when

a Coulomb friction surface is reached:

fµ(tµ) = µ〈tµn〉− + tµsh (23)

where the 〈·〉−

operator returns the negative part of its operand and tµsh =√

(tµs )2 + (tµt )2

is the normalized shear traction. Finally, the friction flow rule is given by:

∆JuKµ= ∆λ

0

tµsh|tµsh|

∆λ =

fµK

(24)

which is complemented by the Kuhn-Tucker conditions λ ≥ 0, fµ ≤ 0, λfµ = 0.

20

4.1.3. Hygrothermal aging

Plasticization and fiber-matrix interface weakening are modeled with an additional

degradation variable dw which is a function of the water concentration c of the material

point:

dw =d∞wc∞

c (25)

where c∞ and d∞w are the water concentration and material degradation at saturation,

respectively. Since only tests at dry and saturated states are available for resin [23] and

interface (Section 2.3.1), a linear evolution of dw is adopted. The degradation variable

is then used to modify the resin properties as follows:

Ew = (1− dw)E (26)

σwc = (1− dw)σc, σw

t = (1− dw)σt (27)

Xwc = (1− dw)Xc, Xw

t = (1− dw)Xt, Gwc = (1− dw)

2Gc (28)

where the squared toughness degradation is adopted for the sake of numerical stability,

since no reliable measurements of saturated fracture toughness on the present resin

system are available and only an estimation of its value based on tensile tests is adopted

[23].

For the fiber-matrix interface, a similar degradation approach is adopted, but the

change in fracture toughness is modified in order to reflect the results obtained on the

fragmentation tests of Section 2.3.1:

Xwn = (1−dw)Xn Xw

sh = (1−dw)Xsh GwIc =

√1− dwGIc Gw

IIc =√1− dwGIIc

(29)

where Xn, GIc, Xsh and GIIc are the interface strength and fracture toughness in mode

I and modes II and III, respectively. Since only mode II properties are available for the

present material system (Table 1), they are also used to describe decohesion in mode I

in the present study.

Previous experiments on neat resin [7] point to complete stiffness and strength re-

covery after a single immersion-redrying cycle at 50 C. Hygrothermal degradation of

21

the resin is therefore modeled as a reversible process. Due to the lack of experimen-

tal data on redried fragmentation specimens, interface degradation is also modeled as

reversible.

4.2. Multiscale/multiphysics framework

The microscopic material models of Section 4.1 are used in conjunction with a

multiscale/multiphysics framework in order to predict the behavior of macroscopic ma-

terial specimens subjected to aging followed by mechanical loads. The framework is

presented in detail in [18] and will be briefly summarized in the following.

4.2.1. Diffusion model

A Fickian diffusion model is used to predict the evolution of the macroscopic water

concentration field with aging time. The changes in concentration and the resultant

mass flux j are computed by:

cΩ +∂

∂xΩ· jΩ = 0 jΩ = −DΩ ∂c

∂xΩ(30)

where D is an orthotropic diffusivity matrix and the superscript Ω indicates macro-

scopic quantities. The water concentration at each material point is downscaled to an

embedded micromodel (FE2) and is considered constant inside the microdomain. This

solution implicitly assumes that water diffusion in the resin is well represented by a

one-phase Fick solution. The diffusivity vector is pre-computed through a homoge-

nization process [34] and is considered constant, leading to a linear transient diffusion

problem.

4.2.2. Stress model

The nonlinear stress problem is solved iteratively using a Newton-Raphson solver

and takes place after the water concentration field is updated (staggered coupling). An

equilibrium problem is solved at both scales:

∂

∂xΩ· σΩ = 0

∂

∂xω· σω = 0 (31)

where the superscript ω indicates microscopic quantities. No assumptions are made

regarding the constitutive behavior of the macroscopic material. Instead, stresses and

22

stiffnesses are obtained through homogenization of the microscopic response: Strains

are applied as prescribed displacements at the corners of the embedded micromodels,

the micro boundary value problems are solved and volume averages of stiffnesses and

stresses are upscaled.

4.3. Reduced-order modeling

The multiscale/multiphysics framework of Section 4.2 involves solving a micro-

scopic boundary value problem for each macroscopic integration point on every itera-

tion of every time step. In turn, solving each micromodel involves solving a nonlinear

finite element equilibrium problem featuring dense meshes and complex material mod-

els. The resultant computational effort is, therefore, exceedingly high. In this section,

two model order reduction techniques will be briefly presented which allow for effi-

ciently solving the microscopic equilibrium problems with minimum loss of accuracy.

For more detailed formulations and an assessment of their efficacy, the reader is re-

ferred to [24].

4.3.1. Proper Orthogonal Decomposition (POD)

Instead of solving for the full micro displacement field uω with N unknowns, one

can instead choose a number n ≪ N of displacement modes arranged in a matrix

Φ ∈ RN×n and solve for their relative contributions α, which reduces the size of

the equilibrium system to n unknowns. The original displacement field is the linear

combination of the n modes:

u = Φα (32)

which leads to reduced versions of the global internal force and tangent stiffness:

fωr = ΦTfω Kωr = ΦTKωΦ (33)

where a subscript r denotes a reduced entity. The displacement modes are computed

before the multiscale analysis by subjecting the micromodel to representative loading

situations and applying a truncated Singular Value Decomposition (SVD) process to

the obtained displacement snapshots [24, 35].

23

4.3.2. Empirical Cubature Method (ECM)

Although POD reduces the size of the global equilibrium equation, computing fω

and Kω still involves computing complex constitutive models for all M integration

points of the microscopic domain ω. Further acceleration (so-called hyper-reduction)

can be achieved by choosing a number m ≪ M of integration points and computing

modified integration weights in order to obtain a good approximation of the reduced

global internal force vector with only a small fraction of the effort:

fω =

∫

ω

f dω ≈m∑

i=1

f (xωi )i (34)

The reduced set of points and weights is chosen by minimizing the error between the

original and approximated versions of fωr using a least-squares point selection algo-

rithm [21, 24].

4.4. Results and discussion

The preceding numerical framework was implemented in an in-house Finite Ele-

ment code built using the open-source Jem/Jive C++ library [36]. The analyses are

executed on a workstation equipped with an Intel Xeon E5-2650 processor and 64Gb

of RAM. Evaluation of the micromodels during concurrent multiscale analyses is per-

formed in parallel. The microscopic domain is represented by a periodic microcell

with 16 fibers. Based on a previously performed RVE study [18], this microdomain

size is considered to represent the macroscopic behavior of the unidirectional compos-

ite material with reasonable accuracy. The micromodel is discretized with 3198 wedge

finite elements and 337 interface elements around the fibers. The fibers are modeled

as linear-elastic, while the material models of Section 4.1 are used for the resin and

interfaces. The adopted microscopic material properties are shown in Table 4.

In order to employ the micromodel in a macroscopic simulation of an ILSS test,

a reduced-order model is first trained to accurately represent the elastic, plastic and

fracture response of a micromodel subjected to a combination of interlaminar shear,

hygrothermal aging and residual stresses. The load cases used in the training process

are shown in Fig. 12. A strain rate ε = 5× 10−2 s−1 is adopted for all cases with

mechanical loads. For cases including water concentration, it is applied in steps from

24

Glass fibers

K∞ [MPa] 43452

G∞ [MPa] 29918

αt [K−1] 5.4 · 10−6

Epoxy resin

K∞ [MPa] 3205

G∞ [MPa] 912

Ku [MPa] 125 182 625 143

Gv [MPa] 36 52 178 41

ku [s] 4.16 · 10−2 2.30 · 100 4.22 · 101 3.11 · 104

gv [s] 1.46 · 10−1 8.08 · 100 1.48 · 102 1.09 · 105

σt [MPa] 64.80− 33.6e−ǫpeq/0.003407 − 10.21e−ǫpeq/0.06493

σc [MPa] 81.0− 42.0e−ǫpeq/0.003407 − 12.77e−ǫpeq/0.06493

ηp [s] 3.49 · 1012

mp [-] 7.305

νp [-] 0.32

Xt [MPa] 83.8− 5.99Υ

Xc [MPa] 104.7− 7.48Υ

Gc [N/mm] 1.9

d∞w [-] 0.14

αt [K−1] 6.0 · 10−5

αsw [%−1] 0.002

Fiber-matrix interface

K [MPa] 106

Xn/Xsh [MPa] 52

GIc/GIIc [N/mm] 0.093

µ [-] 0.4

d∞w [-] 0.42

Diffusion

c∞ [%] 3.2

Dx/Dy [µm/s] 0.520

Dz [µm/s] 1.638

Table 4: Properties used for numerical modeling.

25

Elastic cases

Inelastic cases

Prescribed displacements

Water concentration

Temperature change

Figure 12: Load cases used for training the reduced-order model.

an initially dry state to a concentration value of 3.2%, corresponding to the saturation

level of the epoxy resin. For residual stresses, a temperature variation ∆T = −27 C

is applied in steps. For cases combining environmental loads with mechanical strains,

water is applied first, followed by temperature and finally by prescribed displacements.

Each elastic case is run for three steps in order to provide the reduced model with in-

formation on the decay of viscoelastic stresses. The inelastic cases are run until a stress

drop is observed, indicating strain localization. Since failure occurs due to transverse

shear, it manifests as a single horizontal localization band which runs between the left

and right boundaries of the RVE.

The original model is reduced from N = 12 138 to n = 21 degrees of freedom in

a first reduction stage (POD). This reduced model is then subjected to a second train-

ing stage in order to select a reduced set of integration points, with a reduction from

M = 4546 to m = 589 integration points. Fig. 13 shows the homogenized stress-

strain responses for both the dry and saturated conditions of the full micromodel and

its POD-reduced counterpart (after the first reduction phase). The reduced model is

able to accurately represent the full response of the cases used for training, although

the prediction of the softening response of the saturated material is relatively less accu-

26

0 0.5 1 1.5 2 2.5 3 3.50

10

20

30

40

50

Strain [%]

Sh

ear

stre

ss[M

Pa

]

Dry (reduced)

Dry (full)

Wet (reduced)

Wet (full)

Figure 13: Reduced RVE model response for the inelastic training cases.

rate due to the sharp softening branch caused by the influence of displacement modes

obtained from the dry test. Comparing the reduced internal force vectors for the POD-

reduced and ECM-reduced models, the additional error caused by hyper-reduction is

approximately 1.9%. The resultant speed-ups of the POD-reduced and ECM-reduced

models with respect to the full-order model are 51 and 302, respectively.

Before using the reduced model in a multiscale analysis, it is interesting to as-

sess its performance for non-trained load cases. Full and POD-reduced micromodels

loaded in transverse shear at intermediate water concentration values are executed and

their homogenized responses are compared in Fig. 14. The reduced model is capable

of interpolating from the trained cases and correctly predicting material behavior for

all concentration values, with only small overshoots in failure strain. The largest over-

shoot occurs for a concentration of 1.6%, a case that requires the largest amount of

interpolation from either the dry or saturated states used for training. Influenced by the

displacement modes at failure obtained from the training cases, failure strains are over-

estimated and steep load drops are obtained for all intermediate concentration values. It

is interesting to note that, due to the smaller number of degrees of freedom, the reduced

model features improved numerical robustness. This can be observed, for instance, in

the c = 2.4% curve, a case on which the full analysis is terminated due to lack of con-

27

0 0.5 1 1.5 2 2.5 3 3.50

10

20

30

40

50

Strain [%]

Sh

ear

stre

ss[M

Pa

]

c = 0.8% (reduced)

c = 0.8% (full)

c = 1.6% (reduced)

c = 1.6% (full)

c = 2.4% (reduced)

c = 2.4% (full)

Figure 14: Reduced RVE model response for untrained scenarios of mechanical tests at different levels of

saturation.

vergence as soon as the softening branch of the equilibrium path is reached while the

reduced model maintains the convergence behavior observed during the training cases.

For the next test, the POD-reduced model is assessed on its ability to represent

loading at different strain rates. The full and reduced responses of a micromodel sub-

jected to transverse shear at strain rates both 100 times faster and slower than the one

used for training are shown in Fig. 15. In both cases, the reduced model correctly ex-

trapolates the time-dependent material response in both the elastic and plastic regimes,

while once again showing only a limited loss of accuracy in terms failure strain. As

expected, the reduced model tends to exhibits behaviors closer to the ones used for

training, leading to underestimated values of peak load and failure strain for the very

fast strain rate and overestimated values for the very slow rate. For both studies on

untrained load cases (water concentration and strain rate), the additional error caused

by hyper-reduction (ECM), computed by comparing the norm of the reduced global

internal force vectors with and without ECM, is at most 6%. Both the POD-reduced

and the hyper-reduced models are therefore considered suitable for representing the

full-order model, with significant acceleration and only limited loss of accuracy even

upon considerable extrapolation from the load cases used in training.

28

0 0.5 1 1.5 2 2.5 3 3.50

10

20

30

40

50

Strain [%]

Sh

ear

stre

ss[M

Pa

]

ε = 5.0× 10−4

s−1 (reduced)

ε = 5.0× 10−4

s−1 (full)

ε = 5.0 s−1 (reduced)

ε = 5.0 s−1 (full)

Figure 15: Reduced RVE model response for untrained scenarios of mechanical tests at different strain rates.

A multiscale/multiphysics model of a three-point bending interlaminar shear test is

performed. At the macroscale, symmetry along the x- and z-axes is exploited, and only

one quarter of the short-beam is modeled (Fig. 16). The model is meshed with 8-node

hexagonal elements. It is a known issue that an RVE ceases to exist upon microscopic

strain localization, with the amount of energy dissipated by the macroscopic model

being dependent on the size of the microdomain [37]. In order to bypass this issue, the

thickness of a band of elements close to the specimen center along its thickness, where

strain localization is expected, is made equal to the size of the micromodel [38]. The

complete mesh comprises 1536 elements with a total of 5120 embedded micromodels.

In order to further reduce computational effort, micromodels with a coarser mesh are

used for material points located after the support along the length of the specimen, a

region which is mostly free of stresses. The resultant mesh can be seen in Fig. 17.

The analysis begins with a pre-conditioning phase with a time step ∆t = 25h, during

which a constant water concentration of 3.2% is applied at the boundary. After the

immersion phase, the virtual specimen is cooled down from 50 C to 23 C at a rate

of 5.5 Cmin−1, after which a mechanical test is performed in three-point bending at

1mmmin−1 and the resultant load, distributed over a length of 1.3mm, is used to

compute the apparent interlaminar shear stresses using Eq. (4).

29

2.15mm

10.75mm

5.375mm

Water/50°C

5.375mm

10.75mm

10.75mm

2.15mm5.375mm

5.375mm

x fixed

x

yz

z fixed

y fixed

x

yz

(a) Immersion problem (a) Modeling approach

5.375mm

Figure 16: Multiscale model of the aging and mechanical test of a short-beam specimen.

The multiscale model is used to obtain load-displacement curves for a dry specimen

and specimens immersed for 250 h, 500 h and 1000 h, after which the water concentra-

tion field inside the specimen is non-homogeneous. In these situations, a stand-alone

micromechanical analysis of the point of maximum concentration is not enough to

derive the behavior of the macroscopic specimen, which is instead dictated by stress

redistribution between regions with different water concentrations and suffers the in-

fluence of transient swelling stresses caused by the non-uniform concentration field.

Fig. 18 shows the obtained stress-displacement curves for each condition up until

the point when interlaminar failure occurs at the center of the specimen. Although

the mechanical response for the unaged sample is similar to the one obtained for a

single RVE in pure transverse shear (Fig. 13), the same does not hold after aging.

For an isolated micromodel saturated with water, failure occurs at approximately 34

MPa, which is slightly higher but still reasonably close to the experimentally obtained

strength (Fig. 4). On the other hand, the multiscale model points to a higher peak

load of 39MPa for the saturated specimen. This increase, which can only be captured

in a multiscale analysis, results from the highly non-linear behavior of the saturated

material prior to strain localization (Fig. 13), leading to stress redistribution along the

specimen thickness. Predicting the interlaminar shear response of an aged specimen is

therefore a more complex task than simply computing the micromechanical response

of an isolated material point at its center.

Nevertheless, the gradual degradation behavior with water concentration observed

in Fig. 4 is qualitatively reproduced by the multiscale model (Fig. 19). The numerically

30

c [%]

3.2

2.4

1.6

0.8

0.0

τyz [MPa]

+2.00e+01

+2.50e+00

-1.50e+01

-3.25e+01

-5.00e+01

εpeq [-]

1.60e-02

1.20e-02

8.00e-03

4.00e-03

0.00e-00

Figure 17: Snapshot of the multiscale analysis of a specimen immersed for 250h at the onset of failure.

0 0.05 0.1 0.15 0.20

10

20

30

40

50

Displacement [mm]

Sh

ear

stre

ss(τ

ILSS)

[MPa

]

Unaged 250h W

500h W 1000h W

1500h W 2000h W

500h D 1000h D

Figure 18: Numerically obtained ILSS stress-displacement curves.

0 500 1000 1500 2000

40

42

44

46

48

Immersion time [h]

Pea

kst

ress

(τmax

ILSS)

[MPa

]

Wet

Redried

0

0.2

0.4

0.6

0.8

1

1.2

Wat

erU

pta

ke

[%]

Uptake

Figure 19: Numerical peak loads and water uptake for different immersion times.

31

obtained peak stress for the unaged case is approximately 7MPa lower than the aver-

age value obtained from the experiments. Since failure in the dry specimen is driven

by resin fracture, such discrepancy is not surprising given the fact that no reliable mea-

surement for the fracture toughness of the present epoxy resin is currently available.

Furthermore, the fiber bundles and resin-rich regions seen in Fig. 7 are not modeled,

which would result in differences in crack propagation during the simulation. The high

saturated peak stress levels predicted by the model suggest that interface degradation

is significantly worse than the one used as input (d∞w = 0.42 from Table 4). This in-

dicates that the saturated properties obtained through SFFT are overestimations of the

actual degraded interface performance. Given the large measurement scatter obtained

in those experiments (Section 2.3.1), the adopted values for Xsh and GIIc are estimated

with a high level of uncertainty.

Results on redried micromodels do not show any permanent degradation, with

a peak stress of approximately 49MPa being obtained for both unaged and redried

specimens. This suggests that plastic deformations and interface debonding due to

swelling are not responsible for the observed irreversible degradation, since the effect

of swelling is already captured by the model. These results reinforce the hypothe-

sis of chemical reactions permanently degrading interface performance, a mechanism

also observed in the tomographic scans of Section 3. Including this effect in future

iterations of the present framework is therefore interesting, either as an additional field

problem by modeling the hydrolytic reaction or phenomenologically as a concentration

threshold that assures a minimum level of degradation upon redrying.

As a last numerical example, it is interesting to take advantage of the computational

efficiency of the reduced model in order to investigate the effect of cyclic immersion-

redrying on the transverse shear strength of the composite material. For this example,

the effect of cycling on a point close to the specimen surface is considered, for which

complete saturation is supposed to be reached after 300 s. For simplicity, the mechan-

ical test is performed on the completely dry material (after cycling), eliminating the

need for a concurrent multiscale approach.

Fig. 20 shows the model stress-strain response after 1, 100 and 200 immersion-

drying cycles. Due to the viscous nature of the epoxy model and its dissipation-

32

0 0.5 1 1.5 2 2.5 3 3.50

10

20

30

40

50

Strain [%]

Sh

ear

stre

ss[M

Pa

]

Unaged

1 cycle

100 cycles

200 cycles

Figure 20: Reduced model response for testing after multiple immersion-redrying cycles.

dependent fracture surface, and since saturation and redrying happen at relatively fast

rates, cyclic differential swelling stresses promote a gradual strength decrease. Al-

though it is reasonable to suppose the occurrence of such gradual degradation, further

experimental evidence is required in order to confirm this trend.

5. Conclusions

This work presented a combined experimental and numerical study on hygrother-

mal aging of unidirectional glass/epoxy composites. Macroscopic material degradation

was investigated through mechanical tests on short-beam specimens tested in three-

point bending. Specimens were tested unaged, after having been immersed in water

for various durations and after immersion-redrying cycles. Microscopic hygrothermal

degradation was investigated through mechanical tests on single-fiber fragmentation

specimens and by performing fractographic analyses on aged specimens through X-ray

computed tomography. An attempt was made to numerically reproduce the observed

degradation with a concurrent multiscale/multiphysics finite element model with hyper-

reduced micromodels equipped with a viscous resin model and cohesive fiber-matrix

interfaces with friction.

The mechanical properties of the fiber-matrix interface were estimated through a

33

combination of single-fiber fragmentation tests and reverse modeling. Significant re-

ductions in strength, friction and fracture toughness were observed after aging. How-

ever, the relatively low failure strain of the resin led to a large number of fragments

for which no information could be extracted. Furthermore, the small number of valid

fragments showed a large scatter in debonding behavior.

Results from interlaminar shear tests on specimens aged at 50 C showed a grad-

ual strength decrease which had a strong correlation with the amount of water ab-

sorbed by the specimens. After saturation at 1000 h of immersion, the strength re-

mained unchanged for another 1000 h. Upon redrying, permanent degradation was

observed for immersion times as short as 500 h. Microscopic observations on these

specimens through X-ray computed tomography revealed no signs of widespread in-

terface debonding, in contrast with observations on specimens aged at 65 C. It is

concluded that an osmotic mechanism is activated at the higher temperature, leading

to extensive debonding and additional water uptake. At 50 C, it is hypothesized that

diffusion and reaction are sufficiently slow as to allow the reaction products to leach

out of the specimen before extensive osmosis occurs. In any case, these observations

strongly suggest the occurrence of chemical damage at fiber-matrix interfaces.

The aging phenomenon was numerically simulated in a multiscale/multiphysics

framework. A combination of the POD and ECM reduced-order modeling techniques

was used to construct a reduced model which runs up to 300 times faster than the

full-order one with limited loss of accuracy. The resultant model is able to qualita-

tively capture the experimentally observed dependency of interlaminar shear strength

with water concentration, although uncertainties related to the fracture toughness of the

resin and interface strength and toughness lead to incorrect predictions of both dry and

saturated strengths.

Acknowledgements

The authors acknowledge the contribution of the TKI-WoZ (Grant number TKIW02005)

and IRPWIND projects for motivating and partly funding this research. The authors

are grateful to Catharina Visser, Erik Vogeley, Frank Stroet, Koen Schellevis and Luc

34

Smissaert for the assistance provided in the experimental part of this work.

References

[1] F. Naya, C. Gonzalez, C. S. Lopes, S. van der Veen, F. Pons, Computational

micromechanics of the transverse and shear behaviour of unidirectional fiber re-

inforced polymers including environmental effects, Compos Part A-Appl S.

[2] F. P. van der Meer, Micromechanical validation of a mesomodel for plasticity in

composites, Eur J Mech A-Solid 60 (2016) 58–69.

[3] C. Qian, T. Westphal, R. P. L. Nijssen, Micro-mechanical fatigue modelling of

unidirectional glass fibre reinforced polymer composites, Comput Mater Sci 69

(2013) 62–72.

[4] L. Gautier, B. Mortaigne, B. V., Interface damage study of hydrothermally

aged glass-fibre-reinforced polyester composites, Compos Sci Technol 59 (1999)

2329–2337.

[5] G. M. Odegard, A. Bandyopadhyay, Physical aging of epoxy polymers and their

composites, J Polym Sci Pol Phys 49 (2011) 1695–1716.

[6] R. Polansky, V. Mantlık, P. Prosr, J. Susır, Influence of thermal treatment on

the glass transition temperature of thermosetting epoxy laminate, Polym Test 28

(2009) 428–436.

[7] I. B. C. M. Rocha, S. Raijmaekers, R. P. L. Nijssen, F. P. van der Meer, L. J. Sluys,

Hygrothermal ageing behaviour of a glass/epoxy composite used in wind turbine

blades, Compos Struct 174 (2017) 110–122.

[8] F. Ellyin, C. Rorhbacher, Effect of aqueous environment and temperature on

glass-fibre epoxy resin composites, J Reinf Plast Comp 19 (2000) 1405–1427.

[9] B. Dewimille, A. R. Bunsell, Accelerated ageing of a glass fibre-reinforced epoxy

resin in water, Composites (1983) 35–40.

35

[10] M. C. Lafarie-Frenot, Damage mechanisms induced by cyclic ply-stresses in

carbon-epoxy laminates: Environmental effects, Int J Fatigue 28 (2006) 1202–

1216.

[11] M. Malmstein, A. R. Chambers, J. I. R. Blake, Hygrothermal ageing of plant oil

based marine composites, Compos Struct 101 (2013) 138–143.

[12] A. Gagani, Y. Fan, A. H. Mulian, A. T. Echtermeyer, Micromechanical model-

ing of anisotropic water diffusion in glass fiber epoxy reinforced composites, J

Compos Mater (2017) 1–15.

[13] Y. Joliff, W. Rekik, L. Belec, J. F. Chailan, Study of the moisture/stress effects on

glass fibre/epoxy composite and the impact of the interphase area, Compos Struct

108 (2014) 876–885.

[14] A. Krairi, I. Doghri, A thermodynamically-based constitutive model for thermo-

plastic polymers coupling viscoelasticity, viscoplasticity and ductile damage, Int

J Plas 60 (2014) 163–181.

[15] X. Poulain, A. A. Benzerga, R. K. Goldberg, Finite-strain elasto-viscoplastic be-

havior of an epoxy resin: Experiments and modeling in the glassy regime, Int J

Plas 62 (2014) 138–161.

[16] B. F. Sørensen, Micromechanical model of the single fiber fragmentation test,

Mech Mater 104 (2017) 38–48.

[17] C. Miehe, J. Schotte, J. Schroder, Computational micro-macro transitions and

overall moduli in the analysis of polycrystals at large strains, Comput Mater Sci

16 (1999) 372–382.

[18] I. B. C. M. Rocha, F. P. van der Meer, R. P. L. Nijssen, L. J. Sluys, A multiscale

and multiphysics numerical framework for modelling of hygrothermal ageing in

laminated composites, Int J Numer Meth Eng 112 (2017) 360–379.

[19] K. Terada, M. Kurumatani, Two-scale diffusion-deformation coupling model for

material deterioration involving micro-crack propagation, Int J Numer Meth Eng

83 (2010) 426–451.

36

[20] Z. Liu, M. Bessa, W. K. Liu, Self-consistent clustering analysis: An efficient

multi-scale scheme for inelastic heterogeneous materials, Comput Method Appl

M 306 (2016) 319–341.

[21] J. A. Hernandez, M. A. Caicedo, A. Ferrer, Dimensional hyper-reduction of non-

linear finite element models via empirical cubature, Comput Method Appl M 313

(2017) 687–722.

[22] P. Kerfriden, O. Goury, T. Rabczuk, S. P. A. Bordas, A partitioned model order

reduction approach to rationalise computational expenses in nonlinear fracture

mechanics, Comput Method Appl M 256 (2013) 169–188.

[23] I. B. C. M. Rocha, F. P. van der Meer, S. Raijmaekers, F. Lahuerta, R. P. L.

Nijssen, L. J. Sluys, Numerical/experimental study of the monotonic and cyclic

viscoelastic/viscoplastic/fracture behavior of an epoxy resin, Under Review.

[24] I. B. C. M. Rocha, F. P. van der Meer, L. J. Sluys, Efficient micromechanical

analysis of fiber-reinforced composites subjected to cyclic loading through time

homogenization and reduced-order modeling, Under Review.

[25] ISO 14130 - Fibre-reinforced plastic composites - Determination of apparent in-

terlaminar shear strength by short-beam method, Tech. rep., ISO (1997).

[26] B. W. Kim, J. A. Nairn, Observations of fiber fracture and interfacial debonding

phenomena using the fragmentation test in single fiber composites, J Compos

Mater 36 (2002) 1825–1858.

[27] U. A. Mortensen, T. L. Andersen, J. Christensen, M. A. M. Maduro, Experimental

investigation of process induced strain during cure of epoxy using optical fiber

bragg grating and dielectric analysis, in: In: Proceedings of the 18th European

Conference on Composite Materials, 2018.

[28] F. P. van der Meer, S. Raijmaekers, I. B. C. M. Rocha, Interpreting the single fiber

fragmentation test with numerical simulations, Under Review.

37

[29] L. P. Mikkelsen, [Dataset] A combined experimental/numerical investi-

gation on hygrothermal aging of fiber-reinforced composites, Zenodo -

https://doi.org/10.5281/zenodo.1313247 (2018).

[30] A. Gagani, A. T. Echtermeyer, Fluid diffusion in cracked composite laminates -

Analytical, numerical and experimental study, Compos Sci Technol 160 (2018)

86–96.

[31] A. Arefi, F. P. van der Meer, M. R. Forouzan, M. Silani, Formulation of a con-

sistent pressure-dependent damage model with fracture energy as input, Compos

Struct 201 (2018) 208–216.

[32] Z. P. Bazant, B. Oh, Crack band theory for fracture of concrete 16 (1983) 155–

177.

[33] G. Alfano, E. Sacco, Combining interface damage and friction in a cohesive-zone

model, Int J Numer Meth Eng 68 (2006) 542–582.

[34] I. B. C. M. Rocha, S. Raijmaekers, F. P. van der Meer, R. P. L. Nijssen, H. R. Fis-

cher, L. J. Sluys, Combined experimental/numerical investigation of directional

moisture diffusion in glass/epoxy composites, Compos Sci Technol 151 (2017)

16–24.

[35] P. Kerfriden, P. Gosselet, S. Adhikari, S. P. A. Bordas, Bridging proper orthogo-

nal decomposition methods and augmented newton-krylov algorithms: An adap-

tive model order reduction for highly nonlinear mechanical problems, Comput

Method Appl M 200 (2011) 850–866.

[36] Jive - Software development kit for advanced numerical simulations,

http://jive.dynaflow.com, accessed: 04-03-2018.

[37] V. P. Nguyen, O. Lloberas Valls, M. Stroeven, L. J. Sluys, On the existence of rep-

resentative volumes for softening quasi-brittle materials - a failure zone averaging

scheme, Comput Method Appl M 199 (2010) 45–48.

38

[38] I. M. Gitman, H. Askes, L. J. Sluys, Coupled-volume multi-scale modelling of

quasi-brittle material, Eur J Mech A-Solid 27 (2008) 302–327.

39